The U.S. Supreme Court's ruling on Aereo Inc. purposely kicked a bigger question down the road: What defines a cloud service and when can it be held liable for the way customers use it?

Several technology groups, including the company that oversees the Firefox Web browser, had argued in legal documents that if Aereo were deemed to violate copyright laws, it could mean legal trouble for a broader class of digital services where people also store and access material online, including file-storage service Dropbox Inc. and...